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Duration of urination does not change with body sizePatricia J. Yanga, Jonathan Phama, Jerome Chooa, and David L. Hua,b,1
Schools of aMechanical Engineering and bBiology, Georgia Institute of Technology, Atlanta, GA 30332
Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 14, 2014 (received for review February 6, 2014)
Many urological studies rely on models of animals, such as rats and
generate jets. Instead, they urinate using a series of drops, which
pigs, but their relation to the human urinary system is poorly
is shown by the 0.03-kg lesser dog-faced fruit bat and the 0.3-kg
understood. Here, we elucidate the hydrodynamics of urination
rat in Fig. 2 A–C, respectively.
across five orders of magnitude in body mass. Using high-speed
Fig. 1H shows the urination time for 32 animals across six
videography and flow-rate measurement obtained at Zoo Atlanta,
orders of magnitude of body mass from 0.03 to 8,000 kg. De-
we discover that all mammals above 3 kg in weight empty their
spite this wide range in mass, urination time remains constant,
bladders over nearly constant duration of 21 ± 13 s. This feat is
T = 21 ± 13 s (n = 32), for all animals heavier than 3 kg. This
possible, because larger animals have longer urethras and thus,
invariance is noteworthy, considering that an elephant's blad-
higher gravitational force and higher flow speed. Smaller mam-
der, at 18 L, is nearly 3,600 times larger in volume than a cat's
mals are challenged during urination by high viscous and capillary
bladder at 5 mL. Using the method of least squares, we fit the
forces that limit their urine to single drops. Our findings reveal
data to a clear scaling law shown by the dashed line: T ∼ M0.13(Fig. 1H).
that the urethra is a flow-enhancing device, enabling the urinary
For small animals, urination is a high-speed event of 0.01- to
system to be scaled up by a factor of 3,600 in volume without
2-s duration and therefore, quite different from the behavior of
compromising its function. This study may help to diagnose uri-
the large animals observed that urinate for 21 s. Fig. 1H shows
nary problems in animals as well as inspire the design of scalable
urination time across 11 small animals, including one bat, five
hydrodynamic systems based on those in nature.
rats, and five mice. Their body masses ranged from 0.03 to 0.3 kg.The large error bar for the rats is caused by bladder fullness
urology allometry scaling Bernoulli's principle
varying across individuals. Fig. 2D shows the time course of theurine drop's radius measured by image analysis of high-speed
Medical and veterinary urology often relies on simple, non- video of a rat. To rationalize the striking differences between
invasive methods to characterize the health of the urinary
large and small animals, we turn to mathematical modeling of
system (1, 2). One of the most easily measured characteristics of
the urinary system.
the urinary system is its flow rate (3), changes in which may beused to diagnose urinary problems. The expanding prostates of
Modeling Assumptions. Urination may be simply described math-
aging males may constrict the urethra, decreasing urine flow
ematically. Fig. 1E shows a schematic of the urinary system,
rate (4). Obesity may increase abdominal pressure, causing
consisting of a bladder of volume V and the urethra, which is
incontinence (5). Studies of these ailments and others often
assumed to be a straight vertical pipe of length L and diameter
involve animal subjects of a range of sizes (6). A study of
D. We assume that the urethra has such a thin wall that its in-ternal and external diameters are equal. Urination begins when
urination in zero gravity involved a rat suspended on two legs for
the smooth muscles of the bladder pressurize the urine to P
long periods of time (7), whereas other studies involve mice (8),
measured relative to atmospheric pressure. After an initial tran-
dogs (1), and pigs (9). Despite the wide range of animals used in
sient of duration that depends on the system size, a steady flow of
urological studies, the consequences of body size on urination
speed u is generated.
remain poorly understood.
Previous medical and veterinary studies, particularly cystome-
The bladder serves a number of functions, as reviewed by
trography and ultrasonography, report substantial data on the
Bentley (10). In desert animals, the bladder stores water to be
anatomy, pressure, and flow rate of the urinary system. Fig. 3
retrieved at a time of need. In mammals, the bladder acts as
shows urethral length (8, 15–25) and diameter (15, 24–34), flow
a waterproof reservoir to be emptied at a time of convenience.
rate (35–42), bladder capacity (25, 43–49), and bladder pressure
This control of urine enables animals to keep their homes sanitary
(1, 35, 39, 40, 43, 46, 50) for over 100 individuals across 13 species.
and themselves inconspicuous to predators. Stored urine may evenbe used in defense, which one knows from handling rodents
Several misconceptions in urology have important repercus-
sions in the interpretation of healthy bladder function. For in-
Animals eject fluids for waste elimination, communication, and
stance, several investigators state that urinary flow is driven
defense from predators. These diverse systems all rely on the
entirely by bladder pressure. Consequently, their modeling of the
fundamental principles of fluid mechanics, which we use to
bladder neglects gravitational forces (11–13). Others, such as
predict urination duration across a wide range of mammals. In
Martin and Hillman (14), contend that urinary flow is driven by
this study, we report a mathematical model that clarifies mis-
a combination of both gravity and bladder pressure. In this study,
conceptions in urology and unifies the results from 41 in-
we elucidate the hydrodynamics of urination across animal size,
dependent urological and anatomical studies. The theoretical
showing the effects of gravity increase with increasing body size.
framework presented may be extended to study fluid ejectionfrom animals, a universal phenomenon that has received little
In Vivo Experiments. We filmed the urination of 16 animals andobtained 28 videos of urination from YouTube, listed in
Author contributions: P.J.Y. and D.L.H. designed research; J.P. and J.C. performed re-
. show that urination style is strongly de-
search; P.J.Y. and D.L.H. analyzed data; and P.J.Y. and D.L.H. wrote the paper.
pendent on animal size. Here, we define an animal as large if it is
The authors declare no conflict of interest.
heavier than 3 kg and an animal as small if it is lighter than 1 kg.
This article is a PNAS Direct Submission.
Large animals, from dogs to elephants, produce jets and sheets
1To whom correspondence should be addressed. Email:
of urine, which are shown in Fig. 1 A–D. Small animals, including
This article contains supporting information online at
rodents, bats, and juveniles of many mammalian species, cannot
11932–11937 PNAS August 19, 2014 vol. 111 no. 33

Jetting urination by large animals, including (A) elephant, (B) cow, (C) goat, and (D) dog. Inset of cow is reprinted from the public domain and cited in
. (E) Schematic of the urinary system. (F) Ultrasound image of the bladder and urethra of a female human. The straight arrow indicates theurethra, and the curved arrow indicates the bladder. Reproduced with permission from ref. 20, (Copyright 2005, Radiological Society of North America). (G)Transverse histological sections of the urethra from a female pig. Reproduced with permission from ref. 9, (Copyright 2001, Elsevier). (H) The relationshipbetween body mass and urination time.
Table 1 shows the corresponding allometric relationships to be
This shape factor is nearly constant across species and body
used in numerical predictions for flow rate and urination time.
mass and consistent with the value of 0.17 found by Wheeler
We begin by showing that the urinary system is isometric (i.e.,
et al. (55).
it has constant proportions across animal size). Fig. 3A shows
Peak bladder pressure is difficult to measure in vivo, and in-
the relation between body mass M and urethral dimensions
stead, it is estimated using pressure transducers placed within the
(length L and diameter D). As shown by the nearly parallel
bladders of anesthetized animals. Pressure is measured when the
trends for L and D (L = 35M0.43 and D = 2M0.39), the aspect
bladder is filled to capacity by the injection of fluid. This tech-
ratio of the urethra is 18. Moreover, the exponents are close to
nique yields a nearly constant bladder pressure across animal
the expected isometric scaling of M1/3. Fig. 3B shows the re-
size: Pbladder = 5.2 ± 0.86 kPa (n = 8), which is shown in Fig. 3D.
lationship between body mass and bladder capacity. The bladder's
The constancy of bladder pressure at 5.2 kPa is consistent with
capacity is V ∼ M0.97, and the exponent of near unity indicates
other systems in the body. One prominent example is the re-
spiratory system, which generates pressures of 10 kPa for animals
In ultrasonic imaging (Fig. 1F), the urethra appears circular
spanning from a mosquito to an elephant (56).
(20). However, in histology (Fig. 1G), the urethra is clearlycorrugated, which decreases its cross-sectional area (9). The
Steady-State Equation of Urine Flow. We model the flow as steady
presence of such corrugation has been verified in studies in
state and the urine as an incompressible fluid of density ρ,
which flow is driven through the urethra (51, 52), although the
viscosity μ, and surface tension σ. The energy equation re-
precise shape has been too difficult to measure. We proceed
lates the pressures involved, each of which has units of energy
by using image analysis to measure cross-sectional area A
per cross-sectional area of the urethra per unit length downthe urethra:
from urethral histological diagrams of dead animals in theabsence of flow (9, 53, 54). We define a shape factor α = 4A/
πD2, which relates the urethral cross-sectional area with re-
Pbladder + Pgravity = Pinertia + Pviscosity + Pcapillary:
spect to that of a cylinder of diameter D. Fig. 3C shows theshape factor α = 0.2 ± 0.05 (n = 5) for which the corrected
Each term in Eq. 1 may be written simply by considering the
cross-sectional area is 20% of the original area considered.
pressure difference between the entrance and exit of the
PNAS August 19, 2014 vol. 111 no. 33 11933

Dripping urination by small animals. (A) A
rat's excreted urine drop. (B) A urine drop releasedby the lesser dog-faced fruit bat Cynopterus bra-chyotis. Courtesy of Kenny Breuer and SharonSwartz. (C) A rat's urine drop grows with time. Insetis reprinted from the public domain and cited in . (D) Time course of the drop radii of therat (triangles) along with prediction from our model(blue dotted line, α = 0.5; green solid line, α = 1; reddashed line, α = 0.2).
urethra. The combination of bladder and hydrostatic pressure
our derivations here, however, we assume that the transient
drives urine flow. Bladder pressure Pbladder is a constant given
phase is negligible.
in Fig. 3D. We do not model the time-varying height in thebladder, because bladders vary greatly in shape (57). Thus,
Large Animals Urinate for Constant Duration. Bladder pressure,
hydrostatic pressure scales with urethral length: P
gravity, and inertia are dominant for large animals, which can be
gravity ∼ ρgL,
where g is the acceleration of gravity. Dynamic pressure Pinertia
shown by consideration of the dimensionless groups in
scales as ρu2/2 and is associated with the inertia of the flow.
The viscous pressure drop in a long cylindrical pipe is given by
the Darcy–Weisbach equation (58): Pviscosity = fD(Re)ρLu2/2αD.
αD as the effective diameter of the pipe to keep the cross-
sectional area of the pipe consistent with experiments. TheDarcy friction factor f
The urination time T, the time to completely empty the bladder,
D is a function of the Reynolds number
Re = ρuD/μ, such that f
may be written as the ratio of bladder capacity to time-averaged
D(Re) = 64/Re for laminar flow and
flow rate, T = V/Q. We define the flow rate as Q = uA, where A =
D(Re) = 0.316/Re1/4 for turbulent flow (104 < Re < 105). Drops
generated from an orifice of effective diameter
απD2/4 is the cross-sectional area of the urethra. Using Eq. 4 to
a capillary force (59) of P
substitute for flow speed yields
αD. Substituting these terms
into Eq. 1, we arrive at
By isometry, bladder capacity V ∼ M and urethral length and
The relative magnitudes of the five pressures enumerated in
diameter both scale with M1/3; substitution of these scalings into
Eq. 2 are prescribed by six dimensionless groups, including the
Eq. 5 yields urination time T ∼ M1/6 ≈ M0.16 in the limit of in-
aforementioned Reynolds number and Darcy friction factor
creasing body mass. Agreement between predicted and mea-
and well-known Froude Fr = u= gL and Bond Bo = ρgD2/σ
sured scaling exponents is very good (0.13 compared with 0.16).
numbers (60) as well as dimensionless groups pertaining to the
We, thus, conclude that our scaling has captured the observed
urinary system, the urethra aspect ratio As = D/L, and pressure
invariance in urination time.
ratio Pb = Pbladder/ρgL. Using these definitions, we nondimen-
We go beyond a simple scaling by substituting the measured
sionalize Eq. 2 to arrive at
allometric relationships from Table 1 for L, D, α, V, and Pbladderinto Eq. 5, yielding a numerical prediction for urination time.This prediction (Fig. 1H, solid line) is shown compared with ex-
perimental values (Fig. 1H, dashed line). The general trend is
captured quite well. We note that numerical values are under-predicted by a factor of three across animal masses, likely because
In the following sections, we solve Eq. 3 in the limits of large and
of the angle and cross-section of the urethra in vivo.
small animals.
How can an elephant empty its bladder as quickly as a cat?
In we apply a variation of the Washburn law
Larger animals have longer urethras and therefore, greater hy-
(61) to show that the steady-state model given in Eq. 2 is
drostatic pressure driving flow. Greater pressures lead to higher
accurate for most animals. Animals lighter than 100 kg ach-
flow rates, enabling the substantial bladders of larger animals to
ieve 90% of their flow velocity within 4 s; however, for animals
be emptied in the same duration as those of their much smaller
such as elephants, the transient phase can be substantial. For

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